Minimum bit swaps between given numbers to make their Bitwise OR equal to Bitwise AND

Given two positive integers A and B, the task is to calculate the minimum number of operations required such that Bitwise OR of A and B equals Bitwise AND of A and B are equal i.e (A&B)=(A|B), where, in each operation two indices i and j are chosen and the i^th bit of A is swapped with the j^th bit of B. If it is not possible to make (A&B)=(A|B), print -1.

Examples:

Input: A = 1, B = 2
Output: 2
Explanation:
A₁₀ ≡ 01₂, B₁₀ ≡ 10₂
The following sequence of moves can be performed:

• i = 1, j = 1⇒ A = 11, B = 00 (A|B = 3, A&B = 0).
• i = 1, j = 0⇒ A = 01, B = 01 (A|B = 1, A&B = 1).

Thus, 2 moves are required.

Input: A = 27, B = 5
Output: 3
Explanation:
A₁₀ ≡ 11011₂, B₁₀ ≡ 00101₂
The following sequence of moves can be performed:

• i = 4, j = 3⇒ A = 01011, B = 01101 (A|B = 15, A&B = 9).
• i = 2, j = 2⇒ A = 01111, B = 01001 (A|B = 15, A&B = 9).
• i = 2, j = 1⇒ A = 01011, B = 01011 (A|B = 11, A&B = 11).

Thus, 3 moves are required.

Approach: 
Observation: The main observation to solve this problem is that for (A&B)=(A|B) is that A must be equal to B because if only two bits are set, then only their Bitwise AND and Bitwise OR are equal.

Follow the below steps to solve the problem:

1. Count the number of total set bits in A and B.
2. If the count is odd, the two numbers cannot be made equal, so print -1.
3. Initialize two counters oneZero=0 and zeroOne=0
4. Traverse through the bits of A and B, and do the following:
   • If current bit of A is set and current bit of B is unset i.e (1, 0), increment oneZero.
   • If current bit of A is unset and current bit of B is set i.e (0, 1), increment zeroOne.
5. To minimize the number of operations required, it is optimal to choose two (1, 0) or two (0, 1) indices and swap either one of them, i.e only half of oneZero and zeroOne operations are required.
6. If oneZero is odd(which means zeroOne is also odd), two more operations would be required to turn (0, 1) and a (1, 0) to (1, 1) and (0, 0)
7. So, the final answer is (oneZero/2)+(zeroOne/2)+(oneZero%2?2:0).

Below is the implementation of the above approach: 

C++

// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;
// Function for counting number of set bit
int countSetBits(int n)
{
    int count = 0;
    while (n) {
        n = n & (n - 1);
        count++;
    }
    return count;
}
// Function to return the count of
// minimum operations required
int minOperations(int A, int B)
{
    // cnt to count the number of bits
    // set in A and in B
    int cnt1 = 0, cnt2 = 0;
    cnt1 += countSetBits(A);
    cnt2 += countSetBits(B);

    // if odd numbers of total set bits
    if ((cnt1 + cnt2) % 2 != 0)
        return -1;
    // one_zero = 1 in A and 0 in B at ith bit
    // similarly for zero_one
    int oneZero = 0, zeroOne = 0;
    int ans = 0;

    for (int i = 0; i < max(cnt1, cnt2); i++) {
        int bitpos = 1 << i;
        // When bitpos is set in B, unset in B
        if ((!(bitpos & A)) && (bitpos & B))
            zeroOne++;
        // When bitpos is set in A, unset in B
        if ((bitpos & A) && (!(bitpos & B)))
            oneZero++;
    }
    // number of moves is half of
    // number pairs of each group
    ans = (zeroOne / 2) + (oneZero / 2);
    // odd number pairs
    if (zeroOne % 2 != 0)
        ans += 2;

    return ans;
}

// Driver code
int main()
{

    // Input
    int A = 27, B = 5;

    // Function call to compute the result
    cout << minOperations(A, B);

    return 0;
}

Java

// Java program for the above approach
import java.io.*;
class GFG{

// Function for counting number of set bit
static int countSetBits(int n)
{
    int count = 0;
    while (n != 0) {
        n = n & (n - 1);
        count++;
    }
    return count;
}
// Function to return the count of
// minimum operations required
static int minOperations(int A, int B)
{

    // cnt to count the number of bits
    // set in A and in B
    int cnt1 = 0, cnt2 = 0;
    cnt1 += countSetBits(A);
    cnt2 += countSetBits(B);

    // if odd numbers of total set bits
    if ((cnt1 + cnt2) % 2 != 0)
        return -1;

    // one_zero = 1 in A and 0 in B at ith bit
    // similarly for zero_one
    int oneZero = 0, zeroOne = 0;
    int ans = 0;

    for (int i = 0; i < Math.max(cnt1, cnt2); i++) {
        int bitpos = 1 << i;

        // When bitpos is set in B, unset in B
        if (((bitpos & A) == 0) && ((bitpos & B) != 0))
            zeroOne++;

        // When bitpos is set in A, unset in B
        if (((bitpos & A) != 0) && ((bitpos & B) == 0))
            oneZero++;
    }
    // number of moves is half of
    // number pairs of each group
    ans = (zeroOne / 2) + (oneZero / 2);

    // odd number pairs
    if (zeroOne % 2 != 0)
        ans += 2;

    return ans;
}

// Driver Code
public static void main(String args[])
{

    // Input
    int A = 27, B = 5;

    // Function call to compute the result
    System.out.println( minOperations(A, B));
}
}

// This code is contributed by splevel62.

Python3

# Python3 implementation of the above approach

# Function for counting number of set bit
def countSetBits(n):

    count = 0
    while (n):
        n = n & (n - 1)
        count += 1

    return count

# Function to return the count of
# minimum operations required
def minOperations(A, B):

    # cnt to count the number of bits
    # set in A and in B
    cnt1 = 0
    cnt2 = 0
    cnt1 += countSetBits(A)
    cnt2 += countSetBits(B)

    # If odd numbers of total set bits
    if ((cnt1 + cnt2) % 2 != 0):
        return -1

    # one_zero = 1 in A and 0 in B at ith bit
    # similarly for zero_one
    oneZero = 0
    zeroOne = 0
    ans = 0

    for i in range(max(cnt1, cnt2)):
        bitpos = 1 << i

        # When bitpos is set in B, unset in B
        if ((not(bitpos & A)) and (bitpos & B)):
            zeroOne += 1

        # When bitpos is set in A, unset in B
        if ((bitpos & A) and (not(bitpos & B))):
            oneZero += 1

    # Number of moves is half of
    # number pairs of each group
    ans = (zeroOne // 2) + (oneZero // 2)

    # Odd number pairs
    if (zeroOne % 2 != 0):
        ans += 2

    return ans

# Driver code
if __name__ == '__main__':

    # Input
    A = 27
    B = 5

    # Function call to compute the result
    print(minOperations(A, B))

# This code is contributed by mohit kumar 29

C#

// C# implementation of the above approach
using System;
using System.Collections.Generic;

class GFG{

// Function for counting number of set bit
static int countSetBits(int n)
{
    int count = 0;
    while (n > 0)
    {
        n = n & (n - 1);
        count++;
    }
    return count;
}

// Function to return the count of
// minimum operations required
static int minOperations(int A, int B)
{

    // cnt to count the number of bits
    // set in A and in B
    int cnt1 = 0, cnt2 = 0;
    cnt1 += countSetBits(A);
    cnt2 += countSetBits(B);

    // If odd numbers of total set bits
    if ((cnt1 + cnt2) % 2 != 0)
        return -1;

    // one_zero = 1 in A and 0 in B at ith bit
    // similarly for zero_one
    int oneZero = 0, zeroOne = 0;
    int ans = 0;

    for(int i = 0; i < Math.Max(cnt1, cnt2); i++)
    {
        int bitpos = 1 << i;

        // When bitpos is set in B, unset in B
        if (((bitpos & A) == 0) && (bitpos & B) != 0)
            zeroOne++;

        // When bitpos is set in A, unset in B
        if ((bitpos & A) != 0 && ((bitpos & B) == 0))
            oneZero++;
    }

    // Number of moves is half of
    // number pairs of each group
    ans = (zeroOne / 2) + (oneZero / 2);

    // Odd number pairs
    if (zeroOne % 2 != 0)
        ans += 2;

    return ans;
}

// Driver code
public static void Main()
{

    // Input
    int A = 27, B = 5;

    // Function call to compute the result
    Console.Write(minOperations(A, B));
}
}

// This code is contributed by bgangwar59

Javascript

<script>

// JavaScript implementation of the above approach

// Function for counting number of set bit
function countSetBits(n)
{
    let count = 0;
    while (n) {
        n = n & (n - 1);
        count++;
    }
    return count;
}
// Function to return the count of
// minimum operations required
function minOperations(A, B)
{
    // cnt to count the number of bits
    // set in A and in B
    let cnt1 = 0, cnt2 = 0;
    cnt1 += countSetBits(A);
    cnt2 += countSetBits(B);

    // if odd numbers of total set bits
    if ((cnt1 + cnt2) % 2 != 0)
        return -1;
    // one_zero = 1 in A and 0 in B at ith bit
    // similarly for zero_one
    let oneZero = 0, zeroOne = 0;
    let ans = 0;

    for (let i = 0; i < Math.max(cnt1, cnt2); i++) {
        let bitpos = 1 << i;
        // When bitpos is set in B, unset in B
        if ((!(bitpos & A)) && (bitpos & B))
            zeroOne++;
        // When bitpos is set in A, unset in B
        if ((bitpos & A) && (!(bitpos & B)))
            oneZero++;
    }
    // number of moves is half of
    // number pairs of each group
    ans = parseInt(zeroOne / 2) + parseInt(oneZero / 2);
    // odd number pairs
    if (zeroOne % 2 != 0)
        ans += 2;

    return ans;
}

// Driver code

    // Input
    let A = 27, B = 5;

    // Function call to compute the result
    document.write(minOperations(A, B));

</script>

Output

3

Time Complexity: O(Log₂N)
Auxiliary space: O(1)